Direct digital conversion of baseband signals to super-nyquist frequencies

ABSTRACT

Method and apparatus for shifting of a lowpass signal to a higher frequency and for precompensating distortion caused by a digital to analog converter (DAC). Higher frequency signals are generated by filtering the output of the DAC so that at least one of the replicas generated by the DAC is the signal of interest. In this manner, the higher frequency signals for wireless transmitters, or for other applications, may be generated more easily. Moreover, compensation for distortion caused by the DAC is performed by a precompensator which modifies for distortion, not at the baseband signal, but at a passband frequency.

REFERENCE TO RELATED APPLICATIONS

This application is related to U.S. patent application Ser. No.09/584,431 filed on May 31, 2000 and entitled Precompensation forDistortion due to Digital to Analog Conversion.

BACKGROUND OF THE INVENTION

Lower frequency signals (such as baseband signals) are often convertedto higher frequencies for various applications. One example of aconversion is in the area of wireless transmitters. Typically, abaseband signal is converted to an intermediate frequency (IF) beforebeing converted to the final output radio frequency (RF). In thewireless context, this conversion to an intermediate frequency is oftennecessary because (1) the required filtering may be impossible at thefinal output radio frequency; and/or (2) the use of an IF allows foreasier and more inexpensive filtering and amplification.

Other examples of wherein a signal is translated to a higher frequencyinclude ADSL and cable modems which modulate signals to higherfrequencies.

With recent advances in digital processing speed, it is possible tocreate a digital baseband signal and digitally modulate such a signal,thereby creating a digital passband signal. This digital passbandsignal, sometimes called a digital IF, if it is not at the final outputfrequency, is then sent to a high speed digital to analog converter(DAC) whose output can be subsequently filtered, amplified, and furtherupconverted as desired. Prior art digital implementations perform thismodulation by either multiplying by a sinusoid or by use of a look-uptable containing results of such multiplications.

The constraint for sampling rates in digital systems, that are generallyapplicable, is that the sampling rate should be greater than twice thehighest frequency component in the desired signal. The minimum samplingfrequency is often called the “Nyquist rate.” Prior art implementationsof transmitters have required the sampling rate to be more than twicethe highest frequency component of the modulated signal.

In addition, when a digital signal is converted into an analog signal,distortion may occur. One example of distortion is a multiplication ofthe signal by sin(x)/x. Prior art implementations have attempted tocompensate for this distortion by modifying the baseband signal prior tointroduction to the Digital to Analog Converter (DAC). However,particularly in instances where the baseband signal is being shifted,this implementation may be inadequate since the higher frequencies ofinterest are not the focus of the compensation.

SUMMARY OF THE INVENTION

This invention provides a method of translating a baseband signal tofrequency higher than twice the sampling rate by delivering a lowpasssignal to a DAC and then filtering out all output images except for theimage at the desired frequency.

This invention further provides a novel use of a shifted sin(x)/xresponse to compensate for distortion from the DAC at the frequency ofinterest.

Therefore, an object of the invention is to provide a method andapparatus configured to translate a baseband or lowpass signal to afrequency higher than twice the sampling rate.

It is a further object of the invention to provide a method andapparatus configured to process at least one image which is output ofthe digital to analog converter.

It is also an object of the invention to provide a method and apparatusconfigured to compensate for the distortion from the digital to analogconverter.

It is still a further object of the invention to provide a method andapparatus configured to shift the sin(x)/x response to compensate fordistortion from the DAC at the frequency of interest.

The advantages of the present invention will become apparent to those ofordinary skill in the art by reading the following detailed description,with appropriate reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a prior art implementation ofpostfiltering.

FIG. 2 is a block diagram of a direct conversion of a baseband signal toa passband signal (with the sin(x)/x distortion neglected).

FIG. 3 is a block diagram of a response of a digital to analog converter(DAC) without compensation for distortion.

FIG. 4a is a waveform of the DAC distortion (sin(x)/x response).

FIG. 4b is a waveform of the compensation filter for the baseband signaland the shifted sin(x)/x response.

FIG. 5 is a block diagram of a direct conversion of a baseband signal toa passband signal (with the sin(x)/x distortion precompensated).

FIG. 6a is an example of a baseband signal.

FIG. 6b is an example of a baseband signal, as shown in FIG. 6a, whichis modulated but not precompensated for distortion.

FIG. 6c is a frequency response for a modified baseband signal, as shownin FIG. 6b, after the signal is processed through the DAC.

FIG. 6d are the negative and positive passband signals at theintermediate frequency for FIG. 6c.

DETAILED DESCRIPTION OF THE PRESENTLY PREFERRED EMBODIMENTS

Shifting of a baseband signal to a higher frequency is often animportant task in the transmission of a signal. Prior artimplementations directed to frequency shifting have required thesampling rate to be more than twice the highest frequency component ofthe modulated signal. The present invention, in contrast, generates ahigher frequency signal, and in a preferred embodiment an intermediatefrequency (IF) signal, centered at a frequency above half the Nyquistrate. In this manner, the higher frequency signals for wirelesstransmitters, or for other applications, may be generated more easily.

Compensation for distortion caused by the Digital to Analog Converter(DAC) is likewise an important task. This compensation is typicallyperformed to adjust for distortion in the baseband frequency. Incontrast, in the present invention, the compensation adjusts fordistortion at the frequency of interest. In a preferred embodiment, thefrequency of interest is the IF passband frequency.

The presently preferred embodiments of the invention will now bedescribed by reference to the accompanying figures, wherein likeelements are referred to by like numerals. FIG. 1 is a block diagram ofa prior art implementation of postfiltering. As shown at block 10, thedigital baseband signal is present. In one example, the baseband signalhas frequencies which are in between −f_(s)/4 and f_(s)/4 (where f_(s)is the sampling frequency). The digital baseband signal is convertedinto an analog signal via a Digital to Analog Converter (DAC), as shownat block 12. In a system in which the sampling frequency is f_(s), adigitally sampled signal has the characteristic that the frequencycomponents between −f_(s)/4 and f_(s)/4 are replicated throughout thespectrum from minus infinity to infinity at intervals of the samplingfrequency, as can be seen from the equation for the discrete timeFourier transform of a signal:${x\left( ^{j\quad {wt}} \right)} = {2\pi \quad {\sum\limits_{k = {- \infty}}^{\infty}\quad {\delta \left( {\omega - \omega_{o} - {2\pi \quad k}} \right)}}}$

where ω=2πf. The 2πk term gives rise to the periodic repetition of thebaseband signal at intervals of f_(s).

In prior art applications, these replicas, or images, were consideredundesirable. Hence, a typical system would have a postfilter after theDAC in order to remove the unwanted replicas, as shown at block 14 ofFIG. 1. This postfilter is usually an approximation to a “brick wall”filter which cuts off all frequencies above half the sampling frequency,f>|f_(s)/2|. This eliminates (or severely attenuates) all of the images,keeping only the baseband signal, as shown at block 16 of FIG. 1. Thisbaseband signal can then be modulated in hardware up to the desiredfrequency. The additional hardware necessary to modulate the basebandsignal up to the desired frequency may include (1) heterodyning toincrease the frequency (e.g., an oscillator fed into mixer); or (2)non-linear frequency multiplication (if there are no envelope variationsin the signal) (e.g., diodes or overdriven amplifiers).

In a preferred embodiment of the present invention, the baseband signalmay be modulated up to a higher frequency without the use of a hardwaremodulator. Referring to FIG. 2, there is shown a block diagram of adirect conversion of a baseband signal to a passband signal. Thedistortion associated with the DAC is not considered in this figure, butis discussed subsequently. As shown at block 18 of FIG. 2, a basebandsignal is provided similar to the signal in FIG. 1. An example of thebaseband signal is shown in FIG. 2.

In one example, the baseband signal is situated between −f_(s)/4 andf_(s)/4. Prior to sending the signal to the DAC, the signal is modulatedusing a modulator, as shown at block 20. In one embodiment, themodulator is performed in software by multiplying the baseband signalwith a complex sinusoid having a frequency of f_(s)/4. Themultiplication results in a shifting of the baseband signal to between 0and f_(s)/2. The multiplication also results in a mirror image of thebaseband signal (identical in content) from −f_(s)/2 to 0, as shown inFIG. 2.

Alternatively, in order to shift the baseband and create a mirror image,the fourier transform of the baseband signal may be taken. Thereafter,the transform may be shifted in frequency and the inverse fouriertransform is taken to return to the time domain.

As discussed previously, one method for modulating the signal is bymultiplying by a complex sinusoid. The baseband signal is multiplied bye^(i*2πN(fc/fs)) where N=sample number; f_(c)=carrier frequency(cycles/sec); and f_(s)=sampling frequency. By choosing the propercarrier frequency and sampling frequency, the modulating (by multiplyingby the complex sinusoid) is simplified considerably. For example, if onewishes to have an image centered at 70 MHz, and f_(c)=10 MHz andf_(s)=40 MHz, the baseband signal should be modulated so that thebaseband signal is centered at 10 MHz and −10 MHz. The complex sinusoidto perform this modulation is e^(i*(π/2)*N). Since N is an integer, thevalue of the complex sinusoid is ±1 or ±i. Therefore, multiplication ofthe baseband signal is greatly simplified by merely changing the sign.

The digital low-pass signal is converted to an analog signal by a DAC,as shown at block 22. The output of the DAC is also shown in FIG. 2 withthe baseband signal and images. As discussed previously, the DACgenerates images which ordinarily are discarded. However, in the presentinvention, these images are the signals of interest and the analogbaseband signal is not the signal of interest. Specifically, the imagesat a higher frequency (in one embodiment, the positive and negativepassbands) are the focus of the signal processing. So that, the positiveand negative passbands have the images of the signal (with one of thepassbands having the same signal as the original baseband signal and theother passband having the mirror image of the original baseband signal).Without the use of the modulator prior to conversion with the DAC, theimages located in the negative and positive passbands would be the same(i.e., the same as the original baseband signal).

In addition, the choice of the sampling frequency dictates where theimages are. In particular, the baseband signal can be digitallymodulated to some frequency below f_(s)/2 such that one of the imagesfalls at the desired IF.

Referring to FIG. 2, there is shown a postfilter which receives theoutput of the DAC. The postfilter in a preferred embodiment is apassband postfilter. In the present invention, the postfilter iscentered at the desired IF, rather than being centered at zero as wasdone in the prior art. Thus, low pass signals, including the basebandsignal, and other images are attenuated. And, the intermediate frequencypassband signal is maintained. Thus, the conventional “desired” signal(the baseband signal) is rejected and the conventional “undesirable”signal (at least one, some or all of the images or replicas) are keptfor processing. In a preferred embodiment, the images at the negativeand positive passbands are reserved for processing.

In a preferred embodiment, the postfilter is a surface-acoustic-wave(saw) filter. The saw filter is chosen based on two criteria: the centerfrequency of the filter and bandwidth of the filter around the centerfrequency. One manufacturer for saw filters is Sawtek, Inc. in Orlando,Fla. When using a saw filter, there may be a 15 db loss in the signal ofinterest; if this is the case, an amplifier may be necessary to boostthe signal leaving the saw filter. In an alternate embodiment, thepostfilter is in the form of an L-C filter.

For example, suppose that the sampling rate is 80 MHz and the desired IFis 140 MHz, as shown in FIG. 2. The output of the DAC will have imagescentered at {. . . −140, −100, −60, −20, 20, 60, 100, 140, . . .} MHz,with the images at multiples of −20 MHz being inverses of those atmultiples of +20 MHz. A passband filter of bandwidth 35 MHz centered at140 MHz placed at the output of the DAC would have an output containingonly the 140 MHz IF signal, as shown in FIG. 2.

It should be noted that while the present embodiment is described in thecontext of a transmission system with one or more intermediatefrequencies, the invention is fully applicable to systems which have nointermediate frequency. This invention would have application in anysystem wherein a baseband signal or a low pass signal needs to betranslated to a higher frequency.

Unfortunately, as is well known in the art, a DAC causes distortion inthe converted signal. One example of distortion is in a zero-order holdDAC which has a six(x)/x frequency response, as shown in FIG. 3. Anexample of a baseband signal (block 28) and the resulting distortion ofthe DAC 30 (including distortion to the images) is shown in FIG. 3. FIG.4a shows a waveform of the DAC distortion (sin(x)/x response). FIG. 4afurther shows the location of the IF passband signal, which is thesection of the waveform of interest, and the associated distortion withthe images in the frequency band of interest.

In the prior art, the distortion introduced in a baseband signal whenpassed through a DAC is approximately compensated for with either asimple digital or analog filter that boosts the higher frequencies.While this solves the potential distortion in the baseband signal, thissolution will not correctly compensate for the distortion of higherfrequency images.

To correctly compensate for the distortion, the compensation focuses onthe effect of the distortion on the image (or images) of interest. In apreferred embodiment, the image of interest is at the intermediatefrequency (IF). Therefore, the correct compensation solution for theimage at IF is the inverse of the distortion (with the sin(x)/xdistortion, the correct compensation solution is the inverse of thesin(x)/x response at the IF frequency). In this manner, the filtering ofthe signal prior to conversion by the DAC (and the attendant distortionto the signal that results) allows for a more flat signal in thefrequency spectrum of interest (i.e., a higher frequency spectrum thanbaseband). Moreover, the distortion due to the DAC is “precompensated”(i.e., the distortion is anticipated so that the baseband signal ismodified prior to conversion by the DAC). Therefore, the distortionassociated with the image of interest is minimized due to theprecompensation.

Referring to FIGS. 4b, there is shown one example of the precompensationof the distortion for the intermediate frequency of interest. Theprecompensation is in the form of a waveform for multiplication with thebaseband signal and, in a preferred embodiment, is a shifted sin(x)/xresponse. An example of a baseband signal is shown in FIG. 6a. Thebaseband signal is sent to the precompensator 34, as shown in FIG. 5.The precompensated baseband signal is then sent to a modulator 36.Alternatively, the precompensator may be placed after the modulator sothat the baseband signal is first modulated and then precompensated.Thereafter, the signal is sent to a DAC 38, converting theprecompensated and modulated digital signal into an analog signal withimages. The image (or images) of interest are filtered based on thepassband filter 40 thereby generating a passband signal 42.

In order to determine the waveform for the precompensator, the effect ofthe distortion at the frequency band of interest must be determined. Asdiscussed previously, the digital baseband signal is modulated by themodulator 36 (i.e., shifted and a mirror image generated) prior toconversion from digital to analog. FIG. 6b shows an example of themodulated baseband signal. After sending the modulated baseband signalto the DAC 38, the result (without accounting for distortion) is shownin FIG. 6c. In this case, the frequency band of interest is between3/2f_(s) and 2f_(s), and between −2f_(s) and −3/2f_(s), as shown in FIG.6c. Comparing the signals at the frequency bands of interest (thepositive and negative passbands), it is shown that the negative passbandsignal is the same as the original baseband signal (as shown in FIG. 6a)whereas the positive passband signal is a mirror image of the basebandsignal. Thus, the negative passband signal is chosen as the passband ofinterest for purposes of compensating for the distortion. The distortionat the negative passband signal (the sin(x)/x distortion) is shifted sothat the distortion is centered at zero (baseband). And, the inverse ofthe shifted curve in the baseband frequency band is determined. Fromthat, an impulse response is generated (for example by using an InverseFourier Transform), which gives an impulse response. That impulseresponse is then convolved with the baseband signal in order toprecompensate for the distortion. In the example of FIG. 6d, thepassband of interest is between −2f_(s) and −3/2f_(s), with a center forthe passband being −7/4f_(s). In order to determine the precompensatormultiplier, the center of the passband of interest should be shifted tobaseband (i.e., shifted to the right) and inverted. Thus, in thisexample, the sin(x)/x distortion should be shifted 7/4f_(s) andinverted.

Moreover, in the example previously discussed wherein the IntermediateFrequency (IF) is 140 MHz, the response of the inverse of sin(x)/x (i.e.x/sin(x)) at 140 MHz is shifted so as to be centered at 0 Hz. Also,since the image at 140 MHz is inverted in frequency (as are all evennumbered images), the compensation response must also be inverted infrequency. For odd numbered images, e.g. 100 MHz, this frequencyinversion is not necessary. An equivalent way to understand this is touse the negative IF without inversion for even images.

Suppose, as in the example above, the fourth image is the desired IF.The equation of a baseband compensation filter for this case is:${{H(f)} = \frac{\pi \left( {\frac{f}{fs} - \frac{7}{4}} \right)}{\sin \left\lbrack {\pi \left( {\frac{f}{fs} - \frac{7}{4}} \right)} \right\rbrack}},{{- \quad \frac{fs}{4}}{\langle{f{\langle\frac{fs}{4}}}}}$

The impulse response of this shifted inverted frequency response curve[H(f)] is then convolved with the baseband signal (which is centered at0 Hz) to precompensate the baseband signal prior to being digitallymodulated or sent to the DAC.

Assuming that the baseband signal was flat to begin with, filtering thesignal in this matter will result in an intermediate frequency that hasa flat frequency response.

In the example above, the negative passband was the passband of interest(i.e., the passband where the image is the same as the original basebandsignal). However, the positive passband may be the passband of interestin other instances. For example, if the intermediate passband were inbetween f_(s) and 3/2f_(s) instead of 3/2f_(s) and 2f_(s) as discussedabove, the positive passband would be the passband of interest.Accordingly, the sin(x)/x distortion would be shifted to the left by5/4f_(s) and inverted. And, depending on whether the passband ofinterest is the negative passband or the positive passband, the sin(x)/xdistortion would be shifted either right or left.

Alternatively, the sin(x)/x distortion may in all instances be shiftedeither right or left, and possibly take the mirror image of thedistortion depending on the center of the passband. For instance, thesin(x)/x distortion may always be shifted to the right so that thedistortion of the passband is centered at 0. If the passband is centeredat 3/4f_(s), 7/4f_(s), 11/4f_(s), . . . , then the negative passband isthe passband of interest (the same as the baseband signal) so that noadjustment is necessary. In the example above, the passband was centeredat 7/4f_(s). If the passband is centered at 1/4f_(s), 5/4f_(s),9/4f_(s), . . . , then the positive passband is the passband of interest(the same as the baseband signal), and the mirror image about 0 of thesin(x)/x distortion should be taken.

It is to be understood that additional alternative forms of the variouscomponents of the described embodiments are covered by the full scope ofequivalents of the claimed invention. Those skilled in the art willrecognize that the preferred embodiment described in the specificationmay be altered and modified without departing from the true spirit andscope of the invention as defined in the following claims, whichparticularly point out and distinctly claim the subjects regarded as theinvention.

We claim:
 1. Method of converting a digital lowpass signal to an analogsignal in a frequency spectrum higher than the lowpass signal, themethod comprising the steps of: determining distortion due to convertingthe digital lowpass signal to analog signal in the frequency spectrumhigher than the lowpass signal; modifying the digital lowpass signalbased on the distortion in the frequency spectrum higher than thelowpass signal; converting the digital lowpass signal using a samplingfrequency (f_(s)) to the analog signal, the analog signal having ananalog lowpass signal and replicas, the replicas being at intervals ofthe sampling frequency; selecting at least one of the replicas in thefrequency spectrum higher than the lowpass signal; and rejecting theanalog lowpass signal.
 2. The method of claim 1 wherein the step ofconverting the digital lowpass signal using a sampling frequency to theanalog signal includes using a digital to analog converter.
 3. Themethod of claim 1 wherein the steps of selecting at least one of thereplicas and rejecting the analog lowpass signal include using a filterwhich has a passband in the frequency spectrum higher than baseband. 4.Method of converting a digital lowpass signal to an analog signal in afrequency spectrum higher than the lowpass signal, the method comprisingthe steps of: modulating the digital baseband signal; converting thedigital lowpass signal using a sampling frequency (f_(s)) to the analogsignal, the analog signal having an analog lowpass signal and replicas,the replicas being at intervals of the sampling frequency; selecting atleast one of the replicas in the frequency spectrum higher than thelowpass signal; and rejecting the analog lowpass signal, wherein thedigital lowpass signal is a digital baseband signal and wherein the stepof modulating the digital baseband signal is performed prior to the stepof converting the digital lowpass signal.
 5. The method claim of claim 4wherein the digital baseband signal prior to the step of modulating thedigital baseband signal is between −f_(s)/4 and f_(s)/4 and wherein thedigital baseband signal after the step of modulating the digitalbaseband signal is between −f_(s)/2 and f_(s)/2.
 6. The method claim ofclaim 4 wherein the step of modulating the digital baseband signalincludes: shifting the digital baseband signal between −f_(s)/4 andf_(s)/4 to between 0 and f_(s)/2; and producing a mirror image of theshifted digital baseband signal, the mirror image being between −f_(s)/2and
 0. 7. Method of converting a digital lowpass signal to an analogsignal in a frequency spectrum higher than the lowpass signal, themethod comprising the steps of: compensating for distortion; convertingthe digital lowpass signal using a sampling frequency (f_(s)) to theanalog signal, the analog signal having an analog lowpass signal andreplicas, the replicas being at intervals of the sampling frequency;selecting at least one of the replicas in the frequency spectrum higherthan the lowpass signal; and rejecting the analog lowpass signal,wherein the step of compensating for distortion is performed prior tothe step of converting the digital lowpass signal.
 8. The method ofclaim 7 wherein the step of compensating for distortion includesdetermining the distortion at the frequency spectrum higher than thelowpass frequency and modifying the lowpass signal to compensate fordistortion at the frequency spectrum higher than the lowpass frequency.9. The method of claim 8 wherein the step of converting the digitallowpass signal using a sampling frequency to the analog signal includesusing a digital to analog converter, wherein the digital to analogconverter produces a distortion of a sin(x)/x response, and wherein thestep of modifying the lowpass signal includes convolving the lowpasssignal with an impulse response of the shifted x/sin(x) response. 10.Method of converting a digital baseband signal to an analog signal in apassband frequency, the passband frequency having a higher frequencythan baseband, the method comprising the steps of: modulating thedigital baseband signal to produce a modulated digital baseband signal;converting the modulated baseband signal using a sampling frequency(f_(s)) to the analog signal, the analog signal having an analogbaseband signal and replicas, the replicas being at intervals of thesampling frequency; and filtering out the analog baseband signal and thereplicas except for at least one replica in the passband frequency. 11.The method of claim 10 wherein the modulated digital baseband signalincludes a shifted digital baseband signal and a mirror image of theshifted digital baseband signal.
 12. The method of claim 11 wherein thedigital baseband signal is between −f_(s)/4 and f_(s)/4; and wherein thestep of modulating the digital baseband signal includes shifting thedigital baseband signal from between −f_(s)/4 and f_(s)4 to between 0and f_(s)/2 and producing a mirror image of the shifted digital basebandsignal, the mirror image being between −f_(s)/2 and
 0. 13. The method ofclaim 10 wherein the step of converting the digital baseband signalusing a sampling frequency to the analog signal includes using a digitalto analog converter.
 14. The method of claim 10 wherein the step offiltering out the analog baseband signal and the replicas except for atleast one replica in the passband frequency includes using a passbandpostfilter.
 15. The method of claim 14 wherein the passband postfilterpasses signals in a negative passband and a positive passband.
 16. Themethod of claim 10 further comprising the step of compensating fordistortion prior to the step of converting the shifted digital basebandsignal.
 17. The method of claim 16 wherein the step of compensating fordistortion includes determining the distortion at the passband frequencyand modifying the modulated baseband signal to compensate for distortionat the frequency spectrum higher than the lowpass frequency.
 18. Themethod of claim 17 wherein the step of converting the digital basebandsignal using a sampling frequency to the analog signal includes using adigital to analog converter, wherein the digital to analog converterproduces a distortion of a sin(x)/x response, and wherein the step ofmodifying the lowpass signal includes convolving the lowpass signal withan impulse response of a shifted sin(x)/x response.
 19. Apparatus fortranslating a lowpass signal to higher frequencies, the apparatuscomprising in combination: digital to analog converter having a samplingfrequency of f_(s) and having an input and an output, the inputreceiving a digital lowpass signal and the output having an analoglowpass signal and replicas at intervals of the sampling frequency,wherein the digital to analog converter generates distortion; postfilterhaving an input and an output, the input of the postfilter beingreceived from the output of the digital to analog converter, thepostfilter attenuating the analog baseband signal; precompensator forcompensating the digital lowpass signal based upon the distortion whichis generated by the digital to analog converter, the input for thedigital to analog converter being based on the output of theprecompensator.
 20. The apparatus of claim 19 wherein the postfilterincludes a passband postfilter being centered at a frequency higher thanthe lowpass signal, wherein the digital to analog converter generatesdistortion in the passband, and wherein the precompensator compensatesthe digital lowpass signal based upon the distortion in the passband.21. The apparatus of claim 20 wherein the passband postfilter is at anintermediate frequency, the intermediate frequency being higher than thesampling frequency of the digital to analog converter.
 22. The apparatusof claim 21 wherein the passband postfilter selects at least one of thereplicas and rejects the analog lowpass signal.
 23. Apparatus fortranslating a lowpass signal to higher frequencies, the apparatuscomprising in combination: digital to analog converter having a samplingfrequency of f_(s) and having an input and an output, the inputreceiving a digital lowpass signal and the output having an analoglowpass signal and replicas at intervals of the sampling frequency;postfilter having an input and an output, the input of the postfilterbeing received from the output of the digital to analog converter, thepostfilter attenuating the analog baseband signal; and modulator havingan input and an output, the input of the modulator receiving anunmodified digital lowpass signal, the modulator generating the digitallowpass signal for receipt by the digital to analog converter.
 24. Theapparatus of claim 23 wherein the modulator includes a multiplier, theunmodified digital lowpass signal being multiplied by the multiplier.25. The apparatus of claim 24 wherein the multiplier multiplies theunmodified digital lowpass signal by a complex sinusoid having afrequency f_(s)/4.
 26. The apparatus of claim 23 wherein the digital toanalog converter generates distortion in the passband and furthercomprising a precompensator for compensating the unmodified digitallowpass signal based upon the distortion in the passband which isgenerated by the digital to analog converter.
 27. The apparatus of claim26 wherein the distortion generated by the digital to analog converteris a sin(x)/x response, and wherein the precompensator modifies thelowpass signal by convolving the lowpass signal with an impulse responseof a shifted sin(x)/x response.
 28. Apparatus for translating a digitalbaseband signal to higher frequencies, the apparatus comprising incombination: multiplier having an input and an output, the input of themultiplier receiving the digital baseband signal, the multipliermultiplying the digital baseband signal by a complex sinusoid togenerate a modified digital bandpass signal; digital to analog converterhaving a sampling frequency of f_(s) and having an input and an output,the input receiving the modified digital bandpass signal and the outputhaving an analog baseband signal and replicas at intervals of thesampling frequency; and passband postfilter having an input, the inputof the postfilter being received from the output of the digital toanalog converter, the passband postfilter being centered at a frequencyhigher than the baseband signal.
 29. The apparatus of claim 28 whereinthe frequency of the complex sinusoid is f_(s)/4.
 30. The apparatus ofclaim 29 wherein the passband postfilter is centered at at least one ofthe replicas.